Introduction to Instantons - Rutgers Physics &...

Introductionto Instantons

T. DanielBrennan

QuantumMechanics

QuantumField Theory

Effects ofInstanton-MatterInteractions

Introduction to Instantons

T. Daniel Brennan

February 18, 2015


T. DanielBrennan

QuantumMechanics

QuantumField Theory


1 Quantum Mechanics

2 Quantum Field Theory

3 Effects of Instanton-Matter Interactions


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsPath Integral Formulation of Quantum Mechanics

Quantum mechanics is based around the propagator:

〈xf |e−iHT/~|xi 〉

In path integral formulation of quantum mechanics werelate the propagator to a sum over all possible paths witha phase:

〈xf |e−iHt/~|xi 〉 = N

∫Dx [t] e iS[x]/~


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsWick Rotation

In order to make the integral well defined we perform aWick Rotation:

t → T = −ite−iHt/~ → e−HT/~

S = i

∫dt

1

2

(dx

dt

)2

− V (x)

→ S =

∫dT

[−1

2

(dx

dT

)2

− V (x)

]

= −∫

dT

[1

2

(dx

dt

)2

+ V (x)

]

This is equivalent working with an inverted potential.


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsSemi-Classical Solution

If we are to solve the Classical equations of motion to aninverted potential we arrive at the usual equation:

d2x

dt2− V ′(x) = 0

we also have the conserved “energy” quantity:

E =1

2

(dx

dt

)2

− V (x)

Now assume that x(t) has quantum corrections, we canexpand it in ~. To first order we have the equation:

d2x (1)

dt2− V ′′(x (0))x (1) = 0


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsIntroducing Quantumness

More formally we can perturb around the classical solutionx̄(t):

x(t) = x̄(t) +∑n

cnxn(t)

But now let the xn(t) be eigenvalues of the equation:

d2xndt2

− V ′′(x̄)xn = λnxn

This introduces ”quantumness” to the equations ofmotion. Integrating over all of these perturbations isequivalent to integrating over all paths.Now we want them to be orthogonal so there is no overcounting of perturbations in our integral:∫ T/2

−T/2xn(T ′)xm(T ′)dT ′ = δnm


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsFormal Integration over Modes

Now if we expand the potential around the classicalsolution we get:

V (x) = V (x̄)− (x − x̄)V ′(x̄) +1

2V ′′(x̄)(x − x̄)2 + ...

Now integrating, using the perturbative expansion of x,the first term will give us a factor of e−S0/~ and the lastterm will give us:∏

n

1√λn

=[det(−∂2

t + V ′′(x̄))]−1/2

This is a problem if there exist any λn = 0. But sincethese satisfy the semi-classical equation, these are actuallyjust a deformation of the classical solution so integratingover them corresponds to integrating over the freeparameters of the classical solution.


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsQuick Summary

Consider the Path Integral formulation of QuantumMechanics

Wick Rotate (make time imaginary t → −it), causes us toinvert the potential

Solve the classical equations of motion there, plug intoaction

Integrate over parameter space (zero modes) and multiplyby∏ 1√

λn(quantum fluctuations).


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsWhat are Instantons?

Instantons are classical solutions to the Wick rotatedequations of motion which have non-trivial topology...

To illustrate this, we will work out the canonical doublewell potential.


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsDouble Well

Double well has potential: V (x) = λ(x2 − a2)2

Interested in the classical solutions which are“topologically nontrivial”. In this example that is when aparticle will tunnel from one minima to another.

Each tunneling event goes as ±a tanh(c(t − t ′))

We can glue solutions together in alternating order tohave more general paths. Limit of large separation (quicktunneling): dilute gas approximation.


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsDouble Well (cont.)

Now when we do the path integral we will have anexpression that looks like:

〈a|e−HT/~| − a〉

=N√

det ′(−∂2t + V ′′)

∑n odd

∫ T/2

−T/2dt1...

∫ tn−1

−T/2dtnK

nenS0/~

Since the particle spends most of its time at the bottom ofone of the wells which is approximately harmonic, we canapproximate the normalization by the harmonic oscillator:

N√det ′(−∂2

t + V ′′)≈( ωπ~

)1/2e−ωT/2


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsDouble Well (cont. cont.)

The integration time integration gives us a factor of T n

n! .

Now we get the expression:

〈a|e−HT/~| − a〉 =( ωπ~

)1/2e−ωT/2

∑n odd

T n

n!KnenS0/~

=( ωπ~

)1/2e−ωT/2 1

2

[eKTe

−S0/~ − e−KTe−S0/~

]And similarly:

〈a|e−HT/~|a〉 =( ωπ~

)1/2e−ωT/2

∑n even

T n

n!KnenS0/~

=( ωπ~

)1/2e−ωT/2 1

2

[eKTe

−S0/~ + e−KTe−S0/~

]


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsDouble Well (cont.cont.cont.)

Examining the exponentials we find:

E =~ω2± ~Ke−S0/~

It is now important to determine K. As it turns out, theseare generally very annoying to calculate so I will just tellyou:

K =

(S0

2π~

)1/2( det ′(−∂2t + ω2)

det ′(−∂2t + V ′′(x̄))

)1/2

This is determined by comparing the exact calculation forthe one instanton with the single instanton term in thesum. Furthermore there is a formula for computing thefraction of coefficients:

det ′(−∂2t + ω2)

det ′(−∂2t + V ′′(x̄))

=ψSHO(T/2)

ψ0(T/2)(1)


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum MechanicsUneven Double Well Decay

We can also consider the uneven well. These are picturesof the classical solution and zero mode:

Since there is a node there will actually be a single modewhich has a lower (negative) eigenvalue. This makes Kimaginary.

We now have that the energy has an imaginary part whichis exactly the decay width:

Im[E ] =S0

2π~|K |e−S0/~ = Γ/2


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum Field TheoryDerrick’s Theorem

There are no non-trivial topological solutions to the doublewell’s qft equivalent in dimension other than 2. There areno non-trivial matter solutions due to Derrick’s Theorem.

Take a scalar field theory:

L =1

2(∂µφ)2 − V (φ)

E =

∫dd−1x

[1

2(∂µφ)2 + V (φ)

]= I1 + I2

Assume there is time dependent solution of finite energyφ̄(x). Now define: φ̄λ(x) = φ̄(λx). Then the energychanges as:

Eλ = λ2−d I1 + λ−d I2

Want λ = 1 minimize the energy otherwise the solution isunstable


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum Field TheoryDerricks’ Theorem (cont.)

Varying the Energy with respect to λ we find:

∂Eλ∂λ

∣∣∣λ=1

= (2− d)I1 − d I2 = 0

∂2Eλ∂λ2

∣∣∣λ=1

= (d − 2)(d − 1)I1 + d(d − 1) I2

Solving these we find:

I2 =

(2− d

d

)I1

∂2Eλ∂λ2

∣∣∣λ=1

= −2(d − 2)I1 < 0 for d > 2


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum Field TheoryGauge Instantons

We can however have topological solutions to gauge fields.Take SU(N) gauge theory:

S = −∫

d4x1

2g2(F aµν)2

Require finite energy and finite action. This enforces:

Aµ −→r→∞

U−1∂µU + O

(1

r2

)These can be classified by their winding number at r =∞:

k = − 1

16π2

∫d4x trFµν ∗ Fµν

= − 1

8π2

∫r=∞

d3xnµεµνρσ tr

[Aν∂ρAσ +

2

3AνAρAσ

]= − 1

8π2

∫r=∞

tr

[A ∧ dA +

2

3A ∧ A ∧ A

]


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum Field TheoryWinding Number Classification

This classification holds because:

S = − 1

2g2

∫d4x trF 2

= − 1

4g2

∫d4x tr(F ∓ ∗F )2 ∓ 1

2g2

∫d4 trF ∗ F

≥ ∓ 1

2g2

∫d4x tr(F ∗ F ) =

8π2

g2(±k)

This bound is saturated when F is (anti-)self dual.

These winding numbers are determined by the types ofgauges we can have, that is the class of functions U(x)which maps spatial infinity S3 for Euclidean space to thegauge group. This tells us that the k are exactlydetermined by π3(G ).


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Instantons in Quantum Field TheoryPure Gauge Theory

For a pure gauge theory, everything else is the same asbefore

DeterminantIntegrate over Zero modesClassical action

Now we can use Index Theorems to determine how manyzero modes

Can show exact correspondence and orthogonality of zeromodes from free parameters

Can use Fadeev-Popov method to convert integration overzero modes to integration over parameters


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Effects of Instanton-Matter InteractionsFermions???

When add fermions to theory, everything is the sameexcept for integration over the fermionic zero modes...

There are grassmanian degrees of freedom: N=2 C(R) kIntegration over grassmanian variable picks out zero modesfrom operators in the expectation value of operators

ψ(yi ) = ψi ψ = ψcl +∑n

ψn ψzero ∼ f (x)K

〈O(A, ψ, ψ̄)ψ1...ψN)〉k

= N∫DADψ̄Dψ

N∏i=1

DKi O(A, ψ, ψ̄)ψ1...ψNe−S[A,ψ]

~

∼ 〈O(A, ψ, ψ̄〉0e− 8π2

g2 |k|


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Effects of Instanton-Matter InteractionsBreaking of U(1) Symmetry

Now that we have operators with uneven number offermions and their conjugates, the symmetry of theLagrangian:

ψ → e iϕψ

is broken.If we have a term in Lagrangian:

L =θ

16π2trFµν ∗ Fµν

then this anomalous phase can be compensated by:

θ → θ + 2C (R)kϕ θ ∼ θ + 2π

Now we have a discrete symmetry:

ψ → e iϕψ ϕ =2π

2C (R)k

Important when up in N=2 SUSY Moduli space


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Effects of Instanton-Matter InteractionsBaryon Decay

Consider the Strong Force: SU(3)

The k=1 instanton background leads to an integrationover 6 zero modes. (it removes 6 quark operators fromexpectation values)

This leads to a 6 (anti-)fermion vertex in our effectiveaction

V ∼ u1Lu

2Lu

3Ld

1Ld

2Ld

3L

This clearly violates baryon conservation and can lead toproton, neutron decay:

p + n −→ e+ + ν̄µ

Suppressed by e− 8π2

g2 |k| ∼ 10−34.


T. DanielBrennan

QuantumMechanics

QuantumField Theory


Conclusion

In Quantum Mechanics, instantons describetunneling/decay phenomena by using a semi-classicalapproximation to equations of motion with imaginary time.

In Quantum Field Theory instantons are described bygauge fields with non-trivial winding at infinity. They alsolead to a description of tunneling and decay.

In QFT when instantons interact with matter they giverise to fermionic zero modes which allow for chiralasymmetric operators which lead to U(1) symmetrybreaking and baryon decay.


T. DanielBrennan

QuantumMechanics

QuantumField Theory


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